This paper compares the continuous and discrete adjoint-based automatic aerodynamic optimization. The objective is to study the trade-off between the complexity of the discretization of the adjoint equation for both the continuous and discrete approach, the accuracy of the resulting estimate of the gradient, and its impact on the computational cost to approach an optimum solution. First, this paper presents complete formulations and discretization of the Euler equations, the continuous adjoint equation and its counterpart the discrete adjoint equation. The differences between the continuous and discrete boundary conditions are also explored. Second, the results demonstrate two-dimensional inverse pressure design and drag minimization problems as well as the accuracy of the sensitivity derivatives obtained from continuous and discrete adjoint-based equations compared to finite-difference gradients.
This paper compares the continuous and discrete viscous adjoint-based automatic aerodynamic optimization. The objective is to study the complexity of the discretization of the adjoint equation for both the continuous and discrete approach, the accuracy of the resulting estimate of the gradient, and its impact on the computational cost to approach an optimum solution. First, this paper presents complete formulations and discretizations of the NavierStokes equations, the continuous viscous adjoint equation and its counterpart the discrete viscous adjoint equation. The differences between the continuous and discrete boundary conditions are also explored. Second, the accuracy of the sensitivity derivatives obtained from continuous and discrete adjoint-based equations are compared to complexstep gradients. Third, the adjoint equations and its corresponding boundary conditions are formulated to quantify the influence of geometry modifications on the pressure distribution at an arbitrary remote location within the domain of interest. Finally, applications are presented for inverse, pressure and skin friction drag minimization, and sonic boom minimization problems.
This paper presents an adjoint method for the optimal control of unsteady flows. The goal is to develop the continuous and discrete unsteady adjoint equations and their corresponding boundary conditions for the time accurate method. First, this paper presents the complete formulation of the time dependent optimal design problem. Second, we present the time accurate unsteady continuous and discrete adjoint equations. Third, we present results that demonstrate the application of the theory to a two-dimensional oscillating airfoil. The results are compared to a multipoint approach to illustrate the added benefit of performing full unsteady optimization.
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